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Project 2-Guidelines Recall- Class Project-Goals Determine what would be expected to happen if each company bid the same amount as its signal. Determine the Company 1 bid under several uniform bidding strategies, and explore the expected values of these plans. Find a stable uniform bidding strategy that could be followed by all companies, without any chance for improvement. Recall-Project Assumptions Assumption 1. The same 18 companies will each bid on future similar leases only bidders for the tracts. Assumption 2. The geologists employed by companies equally expert on average, they can estimate the correct values of leases. each signal for the value of an undeveloped tract is an observation of a continuous random variable, Sv, Mean of Sv v ( actual value of lease ) Recall-Project Assumptions Assumption 3. Except for their means, the distributions of the Sv’s are all identical (The shape /The Spread) Assumption 4. All of the companies have the same profit margins Strategies for bidding on an Oil Lease • Strategy 1 -Bid your signal. What will happen? Give reasoning for your analysis • Strategy 2(First Plan) -Subtract Winner’s curse from your signal to obtain bid. Assume all other companies do the same process. What will happen? Give reasoning for your analysis • Strategy 3(Second Plan) -Subtract Winner’s curse and Winner’s blessing from your signal to obtain bid. Assume all other companies do the same process. What will happen? Give reasoning for your analysis Strategies for bidding on an Oil Lease • Strategy 4 -Find a optimal adjustment for company 1. Assume all other companies Subtract Winner’s curse and Winner’s blessing from their signals to obtain their bids. • Strategy 5 -Find a optimal adjustment for company 1. Assume all other companies Subtract Winner’s curse from their signals to obtain their bids. • Strategy 6 -Determine a stable Nash equilibrium bid. This stable strategy is such that any company will not have any incentive to deviate from Strategy 1 -Bid your signal. What will happen? Give reasoning for your analysis • The company (highest signal) will submit the highest bid-> win the lease. • Under this plan, our company would submit a bid of its signal, s1 = $121,600,000 • “Winner’s Extra Profit” is almost always negative. That is, the winning company will not get its needed fair return on the lease. • The mean of the 18 signals for a new lease = the actual value of the lease.Because for example, historical auction 1 the mean lease signal=85.3 M & proven value =91M (Assumption 2) • Hence, the highest signal will almost always be well above the value of the lease and the winning company will have paid too much for the drilling rights. This is called the winner’s curse. • WC=avg of the maximum error column Strategy 2(First Plan) -Subtract Winner’s curse from your signal to obtain bid. Assume all other companies do the same process. What will happen? Give reasoning for your analysis • defeat the winner’s curse. • estimate the expected size of the curse • Let C be the continuous random variable which gives the largest number in a sample of 18 observations of R(error). • E(C)=winner’s curse=avg. of the maximum error column • If a company bids its signal and wins the auction, it can expect, on average, to fall $22,600,000 below its needed fair return on the lease. • If each company bids 22.6 million dollars less than its signal, then any one of the companies is equally likely to win the auction, and there will be no winner’s curse. • Under this plan, our company would submit a bid of 121.6 22.6 million dollars, that is $99,000,000 Strategy 3(Second Plan) -Subtract Winner’s curse and Winner’s blessing from your signal to obtain bid. Assume all other companies do the same process. What will happen? Give reasoning for your analysis • In an auction where the highest bid wins, any amount that the winner pays above the second highest bid is wasted. • Let B be the continuous random variable which gives the difference between the largest and second largest errors in a random set of 18 observations of R. • E(B)=winners blessing=average of the difference column=5.8M • the winning company will, on average, pay an unnecessary premium of 5.8 million. • This leads to another possible bidding strategy, that we will call the Second Plan. • Each company could bid $22,600,000 + $5,800,000 = $28,400,000 less than its signal. • Under this plan, our company would submit a bid of 121.6 28.4 million dollars, that is 93.2M Bidding on Probability, Mathematics, Tests, Homework, Computers an adjustment, The larger the expected value of Company 1’sOil Leasethe better is the long term effect of that bidding plan for us. on the project reduce by both the winner’s curse and blessing for Company 1 and all other companies. each company has approximately the same chance of winning, and, if a company wins, it can expect an extra profit that is close to the 5.8 million dollar winner’s blessing. Results Of Adjustment Average For All Other Company 1 Companies Probability of Winning 0.057 0.055 Mean Extra Profit If Company Wins 5.843 5.832 Expected Value Of Adjustment 0.335 0.323 Auction Focus.xls Simulating, Focus Class Project (material continues) T C I Strategy 4 -Find a optimal adjustment for company 1. Assume all other companies Subtract Winner’s curse and Winner’s blessing from their signals to obtain their bids . • steps • 1. Enter the sum of the WC & WB as the signal adjustment for all other companies cell • 2. Change company 1 signal adjustment cell to get a set of expected values for company 1. • 3. record results of expected value for company 1 • How to find good adjustment points(10 points ) for company1? • For example, If your WC=23,you could use 25,27,29,31,33 • and 21,19,17,15 • 4. enter each good adj. ->hit F9(to recalculate)->manually record the expected values & create a table for company 1 Strategy 4- Bidding on Probability, Mathematics, Tests, Homework, Computers Constructing f(a) function Leasean Oil on the project Company 1 for Company 1 must Signal Adjustment: find the maximum Subtract Expected Value expected value of 13 0.245 adjustment(assuming 15 0.434 that all other 17 0.505 companies subtract 19 0.507 21 0.511 both the curse and 23 0.491 blessing) 25 0.426 this best adjustment, 27 0.373 acb 29 0.327 31 0.237 Strategy 4 f(a) function COMPANY 1: CURSE & BLESSING FOR ALL OTHERS 0.6 0.5 Expected Value 0.4 0.3 0.2 0.1 0.0 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Signal Adjustment copy from the sheet Strategy in Auction Focus.xls. Let f(a) be the expected value for Company 1 for subtracting a million dollars from its signal, assuming that all other companies adjust their signals by both the curse and blessing. Fit a 4th degree polynomial trend line, which we will use as an approximate Computation formula for the unknown function f. a f (a ) f '(a ) Use solver to find the best adjustment 18.9166 0.524 0.000 Strategy 4 • This is the real world of business,competitors may also elect to subtract less than 28.4 million dollars from their signals. • there is a strong incentive for individual companies to deviate from the strategy and subtract less than the curse and blessing. • Company 1’s best adj. 18.9166 million is not itself a stable strategy. Strategy 5 -Find a optimal adjustment for company 1. Assume all other companies Subtract Winner’s curse from their signals to obtain their bids . • steps • 1. Enter the WC as the signal adjustment for all other companies cell • 2. Change company 1 signal adjustment cell to get a set of expected values for company 1. • 3. record results of expected value for company 1 • How to find good adjustment points(10 points ) for company1? • For example,If your WC=23,you could use 25,27,29,31,33 • and 21,19,17,15 • 4. enter each good adj. ->hit F9(to recalculate)->manually record the expected values & create a table for company 1 Strategy 5 Bidding on Probability, Mathematics, Tests, Homework, Computers an Oil Lease Constructing g(a) function on the project Company 1 for Company 1 must Signal Adjustment: Expected Value find the maximum Subtract expected value of 17 -0.190 19 -0.106 adjustment(assuming 21 -0.036 that all other 23 0.007 companies subtract 25 0.020 both the curse and 27 0.033 blessing) 29 0.029 this best adjustment, 31 0.023 33 0.018 ac 35 0.011 37 0.007 Auction Focus.xls Simulating, Focus Class Project T C I (material continues) Strategy 5 g(a) function COMPANY 1: CURSE ONLY FOR ALL OTHERS 0.1 0.1 Expected Value 0.0 -0.1 0 5 10 15 20 25 30 35 40 -0.1 -0.2 -0.2 Signal Adjustment USE the sheet Strategy in Auction Focus.xls. Let g(a) be the expected value for Company 1 for subtracting a million dollars from its signal, assuming that all other companies adjust their signals by curse. Fit a 4th degree polynomial trend line, which we will use as an approximate formula for the unknown function g. Use solver to find the best adjustment the use of Solver in Strategy shows that g(27.5990) = 0. Hence, ac = 27.5990 million dollars. Company 1’s best response to an adjustment of 22.6 million dollars by all other companies is to lower its signal by the considerably larger amount of $27,599,000. Strategy 5 if we know what all other companies plan to do. Moreover, this same information is available to all of the bidders. Need a stable strategy??? If all companies made such a stable adjustment to their signals, then there would be no incentive for anyone to alter the strategy. A stable bidding strategy is also called a Nash equilibrium Strategy 6 How? (a) Use Auction Equilibrium.xls(. (b) FOLLOW THE INSTRUCTIONS IN THIS FILE! (c) Enter appropriate values in cells B10 through E10. (d) Enter a logical value in cell E39. Run the macro Optimize. (the first logical value to use- (2wc+wb)/2 (e) Enter another logical value in cell E39 and press the key F9.. record numbers in a table. (f) See table (g) Find the stable adj for strategy 6 MUST Download new file • http://business.math.arizona.edu/MBD/mbd.html • Link on class webpage first logical value to use for class Strategy 6 project- (2wc+wb)/2=25.5 Company 1 All Other New logical value Optimal Companies Adjustment, amax Adjustment (use 4 decimals) Subtracted From Signal 1 (25+22.5414)/2=23.770 22.5414 25.5 7 2 23.7707 . . 10 Avg of amax=final stable adj for strategy 6 Extra Profit • Extra profit is the amount by which the winning bid is below the fair value of a lease. • Let Xi be the random variable giving the extra profit gained by Company i. The sample of 10,000 simulated auctions is used to approximate E(X1) and the average of E(X2), E(X3), ..., and E(X18) Results Of Adjustment Average Extra profit of winning company For All Other Company 1 Other Companies Company 1 Companies -2.0 -11.0 Probability of Winning 0.123 0.052 0.2 Mean Extra Profit If Company Wins 4.203 5.487 -3.7 Expected Value Of Adjustment 0.517 0.283

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posted: | 9/7/2011 |

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